# PhD opportunities

## Research Environment

The Integrable Systems group has its own seminar series and frequently hosts Workshops and Conferences in Leeds. It has national and international links and collaborative networks. Some of our previous PhD students and post-doctoral researchers contributed lectures in the LMS Classical and Quantum Integrability programme.

Below is a selection of PhD topics proposed in our group. For more information about these, feel free to contact the designated supervisor directly.

## Funding and Admissions Details

Prospective students are welcome to e-mail any staff member of the group to discuss research possibilities. Further details of possible funding and of how to apply can be found on the Postgraduate research opportunities page.

## List of PhD topics:

### Lagrangian multiforms and quantisation of integrable systems

### Supervisors: Dr Vincent Caudrelier

**Summary:** This project aims to study the covariant quantization of integrable field theories. Historically, the most successful approach to quantizing such theories has been through the so-called Quantum Inverse Scattering Method which is based on the notion of quantum R matrix and quantum Yang-Baxter equation. The latter appears as the canonical quantisation (in the sense of Dirac) of the classical Yang-Baxter equation, a key property of the classical r matrix. One of the conceptual drawbacks of canonical quantization is that it breaks (Lorentz) covariance of the underlying spacetime coordinates.

Recently, two independent discoveries were made that open the way to investigate covariant quantization of integrable field theories from different. One is the fact that the classical r matrix has been identified as playing a key role in a covariant version of the Hamiltonian description of certain integrable field theories. The other one is the notion of Lagrangian multiforms which captures integrability in a Lagrangian framework. This offers the possibility of covariant quantization from the perspective of Feynman's path integral. What ties this project together is the beautiful connection made between the above two new approaches in the area of integrable systems - the classical r matrix appears naturally in the context of Lagrangian multiforms. This suggests that such a connection should survive quantization and relate the quantum R matrix with Feynman's path integral formalism.

**Detailed description**:

The discovery of integrable partial differential equations(PDEs) in the pioneering paper by Gardner, Greene, Kruskal and Miura, marked the birth of the modern era of research generally known as {\it integrable systems}. It triggered a myriad of mathematical discoveries (infinite dimensional Hamiltonian systems, infinite dimensional Lie algebras of symmetries, quantum groups...) which led to breakthroughs in classical and quantum physics (soliton dynamics, exact correlation functions in quantum spin chains and quantum field theories, partition functions in gauge theories...). This rich area of Theoretical/Mathematical Physics comprises a large variety of systems whose key feature is that it is possible to ``solve them exactly'', in sharp contrast with usual perturbation and approximate techniques.

(Quantum) field theory is a dominant framework by which one models fundamental laws of nature, such as electromagnetism (Maxwell theory), gravity (general relativity) or fluid dynamics (Navier-Stokes), and tries to predict or understand the occurrence of certain phenomena. Within this vast arena, a large number of {\it integrable} (quantum) field theories have been found over the years. Given their special properties, they provide valuable "theoretical laboratories" where one can obtain explicit and exact results to guide our understanding of intricate phenomena.

Historically, the process of quantizing these integrable models has used the idea of canonical quantization going back to Dirac: in short, one promotes the classical Poisson bracket of observables to a commutator of quantum operators. Beautiful structures have been discovered thanks to this: Poisson-Lie and quantum groups based on the classical and quantum Yang-Baxter equation (the Fields medal rewarded V.G. Drinfel'd for his work on this topic). In turn, this has allowed for tremendous success in computing the all important correlation functions.

From a conceptual point of view, this approach has limitations which have been recognised for decades: the choice of Poisson bracket and Hamiltonian function breaks the natural (covariance) symmetry between the space-time coordinates. The most famous alternative is Feynman's path integral quantization which is based on Lagrangian as opposed to Hamiltonian description of the field theory. Another, less known idea, relies on constructing a covariant Poisson bracket and Hamiltonian at the classical level before trying to quantize canonically.

Because of the overwhelming success of the quantum Yang-Baxter approach, the other two methods just mentioned have received essentially no attention. However, recently two new discoveries have emerged in integrable field theory: 1) a covariant classical $r$-matrix formalism [1,2], 2) a Lagrangian formalism which captures integrability: Lagrangian multiform theory [3,4].

This PhD project proposes to investigate this quantisation problem by taking advantage of the integrability of the model encoded in the multiform. Since multiforms apply equally well for finite or infinite dimensional systems, we expect that one should be able to describe quantum mechanical systems and quantum field theories for which the classical Lagrangian multiform is known. Current examples abound and include: the Toda chain, Gaudin models, the sine-Gordon equation, the modified Korteveg-de Vries equation, the nonlinear Schrödinger equation, Zakharov-Mikhailov models which contain

the Faddeev-Reshetikhin model and recently introduced deformed sigma/Gross-Neveu models as particular cases, etc. Possible tools of investigation can include discretisation of the path integral, as in Feynman's original work, or equivariant localisation techniques which have proved powerful in the exact computation of the partition function in certain (supersymetric) gauge theories. An alternative and complementary direction would be to develop covariant canonical quantisation based on results obtained by the supervisor on covariant Poisson brackets and classical r-matrix structures. Comparison with, as well as guidance from, the well-established theory of the quantum Yang-Baxter equation and quantum groups will be important components of the project.

With this project, you will

1) Discover and learn the rich world of integrable field theories;

2) Contribute to the existing theory of classical integrable field theories and develop new methods to study their covariant quantization using one or both methods described above.

3) Apply your results to compare with existing predictions in certain quantum integrable field theories and, if time allows, investigate the possibility to tackle the very challenging issue of out-of-equilibrium physics using the tools developed in the second step.

Applicants with a strong background in theoretical/mathematical physics are encouraged. Throughout the project, the successful candidate will interact with international research and researchers, attend international conferences, give seminars, and publish the results in international peer-reviewed journals.

Bibliography

[1] V. Caudrelier, M. Stoppato, *A connection between the classical r-matrix formalism and covariant Hamiltonian field theory*, J. Geom. Phys. 148 (2020), 103546.

[2] V. Caudrelier, M. Stoppato, *Multiform description of the AKNS hierarchy and classical r-matrix*, J. Phys. A 54 (2021), 235204.

[3] S. Lobb, F.W. Nijhoff, *Lagrangian multiforms and multidimensional consistency*, J. Phys. A42 (2009), 454013.

[4] D.G. Sleigh, F.W. Nijhoff, V. Caudrelier, *Variational symmetries and Lagrangian multiforms*, Lett. Math. Phys. 110 (2020), 805.

### Quantum Variational Principle and Discrete Integrable Systems

### Supervisor: Professor Frank Nijhoff

**Description**: The project deals with a novel quantum theory that is based on the mathematical structure behind so-called integrable systems, i.e., models that on the classical level are described by certain (mostly nonlinear) differential or difference equations and that exhibit the aspect of `exact solvability' - the fact that they are amenable to exact (rather than approximate or numerical) methods for their solution. Such systems possess a rich mathematical structure (e.g. connections to infinite-dimensional Lie algebras and quantum groups). A key feature is the notion of multidimensional consistency, the property that such models can be extended to compatible systems of equations in spaces of arbitrary dimension. In 2009 Lobb and Nijhoff proposed a novel variational (i.e., least-action) principle that describes this remarkable feature within a Lagrangian formalism.

The new principle forms potentially a paradigm for a new type of fundamental physics.

Recently, the ideas behind this new approach were extended to the quantum realm, and the first steps were taken to formulate a quantum version of this variational principle. The project for a PhD student is to continue this work into the quantum variational principle, elaborating various (integrable) model systems aimed at building a coherent framework together with the development of some new mathematical methodologies.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.

### Elliptic Discrete Integrable Systems

### Supervisor: Professor Frank Nijhoff

**Description**: The term, Integrable Systems refers to a wide class of very special models, described by nonlinear differential (in the continuous case) or difference (in the discrete case) equations possessing a number of remarkable properties. One of the outstanding features is that these equations are exactly solvable in the sense that, rather than having to rely on numerical techniques or approximations, these equations allow for exact (albeit highly nontrivial) methods for their solution. Examples of these are the well-known soliton solutions of certain partial differential equations in this class. In the discrete case, the theory behind these model difference equations has been steadily developing, mostly from the early 1990s onward, together with the mathematical theories which had to be developed alongside (as they were largely non-existent in the discrete case).

The project focuses on elliptic integrable systems, which are those cases where coefficients and generating quantities for these equations are given in terms of elliptic functions. The latter generalizations of trigonometric functions have a rich mathematical structure, some aspects of which are still being explored (e.g. they play a role in Fermat's last theorem), and the integrable models which are defined through them are in a sense at the top of the food chain of models: they form the richest and most general class of equations. To a large extent the solution structure of those elliptic models has yet to be unravelled, and that will form the core of the project. In the project the student will investigate specific examples of such elliptic discrete integrable systems, which will entail not only to try and apply some well-tested techniques to these more complex cases for finding explicit solutions, but also to develop some methods for generating novel examples of such systems. As motivation, these models are expected to have relevance not only for creating novel mathematics, but also potentially for finding new models of fundamental physics.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.